Evolution behaviour of kink breathers and lump- M -solitons ( M → ∞ ) for the (3+1)-dimensional Hirota-Satsuma-Ito-like equation

In this paper, some novel lump solutions and interaction phenomenon between lump and kink M -soliton are investigated. Firstly, we study the evolution and degeneration behaviour of kink breather wave solution with diﬀerent forms for the (3+1)-dimensional Hirota-Satsuma-Ito-like equation by symbolic computation and Hirota bilinear form. In the process of degeneration of breather waves, some novel lump solutions are derived by the limit method. In addition, M -ﬁssionable soliton and the interaction phenomenon between lump solutions and kink M -solitons (lump- M -solitons) are investigated, the theorem and corollary about the conditions for the existence of the interaction phenomenon are given and proved further. The lump- M -solitons with diﬀerent types is studied to illustrate the correctness and availability of the given theorem and corollary, such as lump-cos type, lump-cosh-exponential type, lump-cosh-cos-cosh type. Several three-dimensional ﬁgures are drawn to better depict the nonlinear dynamic behaviours including the oscillation of breather wave, the emergence of lump, the evolution behaviour of ﬁssion and fusion of lump- M -solitons and so on.


Introduction
It is well known that nonlinear partial different equations (PDEs) involved both space and time variables, are fundamental models in nonlinear science, especially in nonlinear dynamics. The wave phenomenon of nonlinear PDEs are playing an essential role in studying various areas of physics and mathematics, such as nonlinear optics, plasmas, oceanography and Bose-Einstein condensates [1][2][3][4][5][6]. Solitons, breathers, lumps and rogue waves are typical wave models to study nonlinear scientific issues [7,8,9].
Lump wave is a special kind of rational function wave with the characteristics of energy concentration and localization property in the space [10][11][12][13][14][15]. Therefore, a series of methods has been quickly developed to obtain lump wave including the first mathematical description of lump wave [16], the long wave limit method [17,18], the direct method [19], the parameter limit method [20], etc. An effective method was e-mail: llxyz891008@163.com (corresponding author) proposed to study the M -lump solutions of the integrable systems in [18], this positive method has attracted a lot of researchers' attention. Many valuable and interesting results have sprung up, lump waves of the (2+1)-dimensional nonlinear equations [21][22][23][24][25][26]. Lump solutions of the (3+1)-dimensional nonlinear systems, such as potential Yu-Toda-Sasa-Fukuyama equation [27], Sharma-Tasso-Olver-like equation [28], B-type Kadomtsev-Petviashvili-Boussinesq equation [29,30]. Lump waves of the (4+1)-dimensional nonlinear equations [20]. Inspired by the above researches, we mainly focus on the (3+1)-dimensional Hirota-Satsuma-Ito-like (HSIl) to study the evolution and degeneration behaviour of breathers and interaction solutions with kink M -solitons in this work. By applying the Horita bilinear method [31,32] and other systematic method, the structures of solutions for this equation are analyzed.
The (3+1)-dimensional Hirota-Satsuma-Ito-like equation is a new nonlinear wave model which is firstly proposed in [33], it is generated from (2+1)-dimensional Hirota-Satsuma-Ito equation [34] and Hirota-Satsuma equation [35], which reflects abundant physical meaning in nonlinear shallow waves. The equation reads where α and β are real nonzero constants, u(x, y, z, t) describes the unidirectional propagation of shallow water waves. The lump solutions are obtained and some interaction phenomenon are discussed in [33]. Based on the previous research, we mainly focuses on investigating evolution and deformation of kink breather waves, the emergence of lump solutions, interaction phenomenon of lump-M -soliton and analyzing dynamical behaviour of each kinds of solutions.
However, to the best of our knowledge, Eq.(1) has many properties and exact solutions that have not been studied, which deserve further study and discussion. The evolution and degradation of breather wave solutions, and the superposition between lump and kink M -soliton (M → ∞) with different kinds has not been studied thoroughly. In this manuscript, we aim to construct breather wave solutions firstly, as degeneration of breather waves, lump waves emerge. Then the interaction between lump and M -soliton solutions of Eq.(1) is given. The structure as follows: In section 2, Based on Hirota bilinear form, we construct kink breather waves with different forms, lump waves will be derived from breather waves by the limit method. In section 3, M -fissionable soliton and sufficient conditions of the existence for interaction between lump and kink M -solitons will be obtained, theoretical proof of superposition behaviour and examples will be given. Finally some conclusions are concluded in the last section.

Evolution and degeneration from breather to Lump solution
In this section, based on the Hirota bilinear method, the breather wave solutions of Eq.(1) with different structural forms are constructed. Under the action of oscillation of breather wave, some lump or lump-type solutions are found out by using parameter limit method [9,21]. Through Painlevé analysis, we assume u(x, y, z, t) = 2(lnf ) x . (2) where f is an unknown real-valued function about x, y, z, t. Eq.(1) can be converted to the bilinear operator D as follows The Hirota bilinear operator D m x D n t are defined by [31] (n, m ≥ 0) On the basis of [8,9], we choose the following test function where δ i k i d i = 0 (i = 1, 2, 3). A multi-breather solitary solution is constructed by inserting Eq.(4)and Eq.(5)into Eq.(2) as follows The solution u expressed in Eq. In order to construct the lump solutions from Eq.(6), the parameters δ i (i = 1, 2, 3) must satisfy: µ, ν are real numbers, m, n, l, r, s are real number or pure imaginary numbers. Letting k 1 → 0, lump solution are given by When k 1 → 0, the characteristics of period and breather of solution Eq.(6) are degenerated as a rational function solution in Eq. (9). Besides, the solution u in Eq.(9) goes to 0 when |x|, |y|, |z|, |t| tend to ∞ in any direction, and shows the characteristic of polynomial attenuation on all spatial variables. Meanwhile, we know that the solution is non-singular as the parameters satisfy (m 2 + l 2 )µ 2 + (m 2 + n 2 )ν 2 > 0 from the expression of Eq.(9). From Fig.2, the lump solution u has a upward peak and a downward valley, the lump solution with this structures is called lump or lump-type solution [12,14].
Case 2: so we get An exact one breather-wave solution of HSIl equation is obtained by combining Eq.(10), (11) and (2), the spatial structure is shown in Fig.3.

Theoretical proof of superposition behaviour
In this section, the lump M -solitons will be investigated, which are also called interaction solutions by some reserchers [34,39,40,41]. and a theorem is given and prove it completely. Now, we assume a test function consisting of quadratic and exponential functions of sum type as follows: δ j e (pj x+qj y+rj z+sj t+σj ) def = Φ(x, y, z, t)+Ψ(x, y, z, t).

Theorem: Assume the function Φ is a solution of Eq.(3), and the parameters satisfy the relations
is also a solution of

Eq.(3).
Proof: By the Hirota bilinear operator D − , we have Thanks to Φ(x, y, z, t) is a solution of Eq.(3) and s j = 0, Ψ t = 0, we have so that According to the form of Φ and Ψ, some relationship of parameters as follows: and where η j = p j x + q j y + r j z + s j t + σ j . Substituting Eq. (18) and (19) into Eq. (17), then the theorem is proved completely.
Corollary: Assume The function Φ(x, y, z, t) satisfies Φ t = 0 and Φ xxx + αΦ y + βΦ z = 0, and the parameters satisfy the is also a solution of Eq.(3).

Superposition behaviour of Lump-M -soliton
In this subsection, we will give some examples to illustrate the effectiveness of above theorem and corollary, the evolution behaviour of spatio-temporal structure of lump-M -solitons with the change of soliton number M and time t will be studied.
When N = 2, M = 0 in Eq. (14), substituting Eq.(14) into Eq. (3), we obtain the following two solutions with the aid of Mathematica: Firstly, we give the application of the theorem, the spatio-temporal evolution behaviour of lump-Msolitons with the change of soliton number M by drawing three-dimensional plots (Fig.6,7    Secondly, we investigate the application of the corollary. Actually, there are so many functions Φ(x, y, z, t) that satisfy Φ t = 0 and Φ xxx + αΦ y + βΦ z = 0, we will give two examples as follows: Example 1: We study a new interaction phenomenon, Combining Eq.(23) and the corollary. Here, the test function is written as Substituting Eq.(26) into Eq.(2), interaction solutions of lump-L-cosh (L → +∞) type are obtained. In order to better demonstrate the behaviour of spatial structure with the increase of the number of cosh type L. In Fig.9, the lump generates from the kink wave and it is gradually drowned or swallowed by the kink wave. When t = 0, the amplitude gets to the extreme point, the collision between a lump and 1-cosh kink wave is completely nonelastic. The fusion process shown in Fig.10, the lump generates from the of the intersection of 2-cosh kink wave, then it vanishes ad the amplitude changes rapidly. This phenomenon is similar to the rogue wave, it means a kind of rogue wave will appear based on the interaction between a lump and 2-cosh kink wave. The lump generates from the of the intersection of 3-cosh kink wave in Fig.11.

Conclusion
Based on symbolic computation and Hirota bilinear method, the evolution and degradation behaviours of breather wave solutions with different kinds are investigated, some new lump and lump-type solutions are obtained from the breather wave solutions via using the parameter limit method. Besides, inspired by the previous researches on the study of interaction phenomenon, we have studied the superposition behaviour between lump solution and different types of M -solitons (M → ∞) for (3+1)-dimensional HSIl equation, the theorem proof has been proposed firstly in this paper. Meanwhile, several threedimensional spatio-temporal structure figures of breather waves and the interaction solutions are drawn to better reflect the oscillation of solitary wave and the evolution of interaction behaviour, including the oscillation and fusion-fission phenomenon, superposition and evolution behaviour of lump-M -solitons and so on. Fig.1 and Fig.3 show the oscillation of kink breather wave including double breather and breather solitary wave. In Fig.2 and Fig.4, lump solutions are generated from kink breather wave, it is also kink lump waves. M -soliton solutions are known as M -fissionable wave which are described in Fig.5. Some diverse interaction phenomenon shown in the following Figures have great significance to the nonlinear waves in fulid mechanics. In this paper, the methods used to obtain new exact solutions also can be extended to solve other nonlinear partial different equations. For some important classical mathematical and physical models, the methods of studying solutions are very significant.